Luật mạnh số lớn đối với dãy phần tử ngẫu nhiên trên không gian tuyến tính định chuẩn

(Ω, F, P) G σ
σ F X : Ω → R G
B ∈ B(R) X
−1
(B) ∈ G
X F
X
G
X
σ(X) =

X
−1
(B) : B ∈ B(R)

σ σ F σ σ
X σ X X
G σ(X) ⊂ G
X
1
, X
2
, , X
n
(Ω, F, P) f : R

n
→ R ( B(R
n
)/B(R) )
Y = f(X
1
, , X
n
) : Ω → R
ω → f(X
1
(ω), , X
n
(ω))
X, Y (Ω, F, P )
f : R → R a ∈ R
aX, X±Y, XY, |X|, f(X), X
+
= max(X, 0), X
−
= max(−X, 0), X/Y (Y = 0)
{X
n
, n  1}
(Ω, F, P) inf
n
X
n
sup
n

X
n
inf
n
X
n
, sup
n
X
n
, limX
n
, limX
n
lim
n→∞
X
n
( )
X
{X
n
, n  1} X
n
↑ X n → ∞
(Ω, F, P) X : Ω → R
P
X
: B(R) → R
B → P

X
(B) = P(X
−1
(B))
X
P
X
B(R)
Q B(R) Q
X
(Ω, F, P) X : Ω → R
F
X
(x) = P(X < x) = P(ω : X(ω) < x)
X
F
X
(x) = P

X
−1
(−∞, x)

= P
X
[(−∞, x)]
0  F(x)  1.
a < b F (b) − F(a) = P(a  X < b) F (x)
lim
x→+∞

F (x) = 1, lim
x→−∞
F (x) = 0
lim
x↑a
F (x) = F (a) lim
x↓a
F (x) = P(X  a). F (x)
F (x) a P(a) = 0.
F (+∞) = lim
x→+∞
F (x), F (−∞) = lim
x→−∞
F (x).
F (+∞) = 1 F (−∞) = 0.
X : (Ω, F, P) → (R, B(R))
X P
X EX
EX =

Ω
XdP.
E|X|
p
< ∞ (p > 0) X p
E|X| < ∞ X
X  0 EX  0.
X = C EX = C
EX C ∈ R E(CX) = CEX.
EX EY E(X ± Y ) = EX ± EY.

X  0 EX = 0 X = 0.
X
n
↑ X ( X
n
↓ X)
n EX
−
n
< ∞ ( EX
+
n
< ∞) EX
n
↑ EX (
EX
n
↓ EX)
X
n
 Y n  1 EY > −∞
ElimX
n
 limEX
n
.
X
n
 Y n  1 EY < +∞
ElimX

n
 limEX
n
.
|X
n
|  Y n  1 EY < ∞
ElimX
n
 limEX
n
 limEX
n
 ElimX
n
.
|X
n
|  Y n  1
EY < ∞ X
n
→ X X E|X
n
− X| → 0 EX
n
→ EX
n → ∞
X
ε > 0
P(X  ε) 

EX
ε
.
X DX := E(X −
EX)
2
X
X
X
DX =






(x
i
− EX)
2
p
i
X P(X = x
i
) = p
i
;

+∞
−∞

(x − EX)
2
p(x)dx X p(x).
DX = EX
2
− (EX)
2
DX  0
DX = 0 X = EX = C ( ) h. c. c.
D(cX) = c
2
DX
X
DX ε > 0
P(|X − EX|  ε) 
DX
ε
2
.
p > 0 L
p
= L
p
(Ω, F, P)
(Ω, F, P)) E|X|
p
< ∞ X ∈ L
p
, p  1
X

p
= (E|X|
p
)
1/p
.
p
X, Y ∈ L
2
E|XY |  X
2
Y 
2
.
p, q ∈ (1; +∞)
1
p
+
1
q
= 1
X ∈ L
p
, Y ∈ L
q
E|XY |  X
p
Y 
q
.

X, Y ∈ L
p
, 1  p < ∞
X + Y ∈ L
p
X + Y 
p
 X
p
+ Y 
p
.
0  p < 1
C
r
X, Y ∈ L
r
, r > 0
E|X + Y |
r
 c
r
(E|X|
r
+ E|Y |
r
),
c
r
= max(1, 2

r−1
)
ϕ : R → R
ϕ(X)
Eϕ(X)  ϕ(EX).
X ∈ L
t
0 < s < t
X
s
 X
t
.
A B
P(AB) = P(A)P(B).
{A
i
, i ∈ I}
{A
i
, i ∈ I}
A
i
1
, A
i
2
, . . . , A
i
n

P(A
i
1
A
i
2
. . . A
i
n
) = P(A
i
1
)P(A
i
2
) P(A
i
n
).
(Ω, F, P)
{C
i
: i ∈ I, C
i
⊂ F} ( )
A
i
∈ C
i
{A

i
, i ∈ I}
{X
i
, i ∈ I} ( )
σ {σ(X
i
), i ∈ I}
( )
( )
{X
i
, i ∈ I} f
i
: R → R(i ∈ I)
{f
i
(X
i
), i ∈ I}
{X
n
, n  1}
n  1 σ(X
k
, 1  k  n) σ(X
k
, k  n + 1)
X
1

, X
2
, , X
n
F
X
1
,X
2
, ,X
n
(x
1
, x
2
, , x
n
) = P(X
1
< x
1
, X
2
< x
2
, , X
n
< x
n
)

(x
i
∈ R, i = 1, n).
X
1
, X
2
, , X
n
F
X
1
,X
2
, ,X
n
(x
1
, x
2
, , x
n
) = F
X
1
(x
1
)F
X
2

(x
2
) F
X
n
(x
n
).
X Y
E(XY ) = EXEY.
X
1
, X
2
, . . . , X
n
E(X
1
X
2
. . . X
n
) = EX
1
EX
2
. . . EX
n
.
X Y

D(X ± Y ) = DX + DY
X
1
, X
2
, , X
n
D(X
1
+ · · · + X
n
) = DX
1
+ · · · + DX
n
.
{X, X
n
, n  1}
(Ω, F, P)
• {X
n
, n  1} X n → ∞
N ∈ F P(N) = 0 X
n
(ω) → X(ω) n → ∞
ω ∈ Ω\N
X
n
→ X X

n
h. c. c.
−−−→ X n → ∞
• {X
n
, n  1} X n → ∞ ε > 0
∞

n=1
P(|X
n
− X| > ε) < ∞.
X
n
c
−→ X n → ∞
• {X
n
, n  1} X n → ∞
ε > 0
lim
n→∞
P(|X
n
− X| > ε) = 0.
X
n
P
−→ X n → ∞
• {X

n
, n  1} p > 0 X n → ∞
X, X
n
(n  1) p lim
n→∞
E|X
n
− X|
p
= 0
X
n
L
p
−→ X n → ∞
• {X
n
, n  1} ( ) X n → ∞
lim
n→∞
F
n
(x) = F (x) x ∈ C(F ).
F
n
(x) F (x)
X
n
X C(F ) F (x)

X
n
D
−→ X
p L
p
X
n
h. c. c
−−−→ X ε > 0
lim
n→∞
P(sup
mn
|X
m
− X| > ε) = 0.
X
n
c
−→ X X
n
h. c. c
−−−→ X
∞

n=1
E|X
n
− X|

p
< ∞ p > 0 X
n
h. c. c
−−−→ X.
{X
n
, n  1} X
n
h. c. c
−−−→ C
X
n
c
−→ C
X
n
h. c. c
−−−→ X X
n
L
r
−→ X p > 0
X
n
P
−→ X.
X
n
P

−→ X X
n
D
−→ X.
X
n
D
−→ X P(X = C) = 1 X
n
P
−→ X.
{X
n
, n  1}
• P( lim
m,n→∞
|X
m
− X
n
| = 0) = 1
• lim
m,n→∞
P(|X
m
− X
n
| > ε) = 0 ε > 0
• p > 0 lim
m,n→∞

E|X
m
− X
n
|
p
= 0
{X
n
, n  1}
{X
n
, n  1}
{X
n
, n  1}
lim
n→∞
P( sup
k,ln
|X
k
− X
l
| > ε) = 0 ε > 0
lim
n→∞
P(sup
kn
|X

k
− X
n
| > ε) = 0 ε > 0.
{X
n
, n  1}
{X
n
k
, k  1} ⊂ {X
n
, n  1} {X
n
k
, k  1}
{X
n
, n  1}
{X
n
, n  1}
{X
n
, n  1}
{X
n
k
, k  1} ⊂ {X
n

, n  1} {X
n
k
, k  1}
p  1 {X
n
, n  1} p
{X
n
, n  1} p
X
1
, X
2
, . . . , X
n
EX
i
= 0 DX
i
= σ
2
i
i = 1, 2, . . . , n S
k
= X
1
+ + X
k
1  k  n ε > 0

P(max
1kn
|S
k
|  ε) 
1
ε
2
n

i=1
σ
2
i
.
P(max
1kn
|X
k
|  c) = 1
P( max
1kn
|S
k
|  ε)  1 −
(ε + c)
2

n
i=1

σ
2
i
.
{X
n
, n  1}
2 ε > 0
P

max
nmk
|S
m
− S
n
| > ε


1
ε
2
k

m=n+1
EX
2
m
P


sup
mn
|S
m
− S
n
| > ε


1
ε
2
∞

m=n+1
EX
2
m
.
{X
n
, n  1}
2
E

max
1kn




k

i=1
X
i



2

 2
n

i=1
EX
2
i
.
{X
n
, n  1}
EX
i
= a
i
(i = 1, 2, . . . )
• {X
n
, n  1}
X

1
+ · · · + X
n
n
−
a
1
+ · · · + a
n
n
P
−→ 0 n → ∞.
• {X
n
, n  1}
{b
n
, n  1} 0 < b
n
↑ ∞
X
1
+ · · · + X
n
b
n
−
a
1
+ · · · + a

n
b
n
P
−→ 0 n → ∞.
{X
n
, n  1} (
)
{X
n
, n  1}
1
n
2
n

i=1
DX
i
→ 0 n → ∞.
{X
n
, n  1}
{X
n
, n  1}
0 < b
n
↑ ∞

∞

n=1
DX
n
b
2
n
< ∞
1
b
n
n

k=1
(X
k
− EX
k
) → 0 h. c. c.
{X
n
, n  1} sup
n
DX
n
=
C < +∞
1
n

n

i=1
(X
i
− EX
i
) → 0 (n → ∞).
{X
n
, n  1}
E|X
1
| < ∞
1
n
n

i=1
X
i
→ EX
1
h. c. c.
{X
n
, n  1}
1
n
n


i=1
X
i
C E|X
1
| < ∞
C = EX
1
{X
n
, n  1}
E|X
n
| < ∞
EX
n
= a ( ) n ∈ N {X
n
, n  1}
X
1
+ X
2
+ · · · + X
n
n
P
−→ a n → ∞.
f

n
( ) n → ∞
(Ω, F, P) (M, d)
d M B(M) σ M σ
M
V : Ω → M
M {ω ∈ Ω : V (ω) ∈ B} ∈ F B ∈ B(M).
V : Ω → M
M F/B(M)
V : Ω → M
|V (Ω)| |V (Ω)| V
|V (Ω)|
V (Ω)
V M
T : M → M
1
M M
1
σ T (V ) M
1
.
{E
n
, n  1} ⊂ F
∪
∞
n=1
E
n
= Ω. {x

n
, n  1} M V
Ω M V (ω) = x
n
ω ∈ E
n
. V
M.
B ∈ B(M) {x
n
j
, j  1}
{x
n
, n  1} B V
−1
(B) = ∪
∞
j=1
E
n
j
∈ A. V
{V
n
, n  1}
M V
n
(ω) → V (ω) ω ∈ Ω. V
V

−1
(C) ∈ F
C M. k C
k
= ∪
x∈C
N(x,
1
k
),
N(x,
1
k
) = {y ∈ M : d(x, y) <
1
k
}.
C
k
V
−1
(C) =
∞

k=1
∞

n=1
∞


m=n
V
−1
m
(C
k
).
m V
m
M C
k
V
−1
m
(C
k
) ∈ F. V
−1
(C) ∈ F.
M
V : Ω → M
V.
V V
V : Ω → M {x
n
, n  1}
M.
n  1
L
1

= S(x
1
; 1/n);
L
2
= S(x
2
; 1/n)L
1
;
. . .
L
m
= S(x
m
; 1/n)
m−1

k=1
L
k
;
. . .
L
i
∩ L
j
= ∅ i = j)
L
m

∈ B(M)
{x
n
, n  1} M =
∞

m=1
L
m
J = {m : L
m
= ∅}
m ∈ J y
m
∈ L
m
T
n
: M → M
T
n
=

m∈J
y
m
I
L
m
( T

n
(x) = y
m
x ∈ L
m
).
T
n
B(M)/B(M)
B ∈ B(M)
T
−1
n
(B) =

{i : y
i
∈B}
L
i
∈ B(M).
V
n
= T
n
◦V |V
n
(Ω)| = |T
n
(V (Ω))|  |T

n
(M)| = |J| |V
n
(Ω)|
V
n
ω ∈ Ω V (ω) ∈ M
m V (ω) ∈ L
m
d(V
n
(ω), V (ω)) = d(T
n
(V (ω)), V (ω)) <
2
n
.
sup
ω∈Ω
d(V
n
(ω), X(ω)) <
2
n
→ 0
n → ∞
(M, d) U, V
M d(U, V )
B(M) × B(M) σ E × F
E, F ∈ B(M). B(M × M) = B(M) × B(M)

(U, V ) : Ω → M × M (U, V )(ω) = (U(ω), V (ω)) ω ∈ Ω
M × M. d : M × M → R
d(U, V )
R. d(U, V )
{V
n
, n  1}
M {V
n
, n  1}
V
P

lim
n→∞
d(V
n
, V ) = 0

= P

ω : lim
n→∞
d(V
n
(ω), V (ω)) = 0

= 1.
V
n

h.c.c
−−→ V n → ∞
ε > 0
lim
n→∞
P [d(V
n
, V ) > ε] = lim
n→∞
P [ω : d(V
n
(ω), V (ω)) > ε] = 0.
V
n
P
−→ V n → ∞
r(r > 0)
lim
n→∞
E [d(V
n
, V )
r
] = 0.
V
n
L
r
−→ V n → ∞
ε > 0

∞

n=1
P [d(V
n
, V ) > ε] < ∞.
V
n
c
−→ V n → ∞
{V
n
, n 
1} r
V n → ∞
{d(V
n
, V ), n  1}